University of LjubljanaFaculty of Mathematics and Physics

Seminar 4. letnik

Onsager theory of hydrodynamicturbulence

Author: Milos BajicAdvisor: prof. dr. Rudolf Podgornik

Ljubljana, April 2, 2010

Abstract

The goal of this seminar is a brief summary of Onsager’s published and unpublishedcontributions to hydrodynamic turbulence and an account of their place in the field asthe subject has evolved through the years, but main focus will be on two-dimensionalfluids, point-vortices, Saturns “stable” Great red spot explained using term of nega-tive absolute temperature and sophisticated method using non-point-vortex model butrather local distribution function, which is more realistic behavior of vortices.

Contents

1 Introduction 3

2 Brief mathematical review of hydrodynamics 4

3 Onsager theory of turbulence 63.1 Onsager’s theory of point-vortex equilibria . . . . . . . . . . . . . . . . . . . 63.2 Point-vortex model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Advance and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Conclusions 13

References 14

1 INTRODUCTION 3 of 14

1 Introduction

Lars Onsager, born in Oslo, Norway, a science genius, won a 1968 Nobel Laureate inChemistry. He made deep contributions to many ares of physics and chemistry are widelyappreciated. His huge contribution to the thermodynamics of irreversible processes and akey result, the reciprocal relation for linear transport coefficient. Among the other cele-brated contributions are his work on liquid helium, including quantization of circulation,his semiclassical theory of the de Haas-van Alpehn effect in metals 1, his entropic theory oftransition to nematic order for rod-shaped colloids, and the reaction field in his theory ofdielectrics.

Probably less known among physicists is Onsager’s interest in hydrodynamic turbulence.He published two papers on the subject of fully developed turbulence (see [1],[7]). 1945,Onsager predicted an energy spectrum for velocity fluctuations that rolls off as the −5/3power of the wave number. The published abstract appeared a few years after, but entirelyindependently of, the now-famous trilogy of papers by Kolmogorov, proposing his similaritytheory of turbulence. His only full-length article on the subject in 1949 introduced twoideas - negative-temperature equilibria for two-dimensional ideal fluids and energy dissipationanomaly for singular Euler solutions - that stimulated much later work. Reamarkably,his private notes of the 1940s contain the essential elements of at least four major resultsthat appeared decades later in the literature: a mean-field Poisson-Boltzmann equation andother thermodynamic relations for point vortices; a relation similar to Kolmogorov’s 4/5 lawconnecting singularities and dissipation . . .

1often abbreviated dHvA, was descovered 1930; dHvA effect is quantum mechanical effect in the mag-netic moment of a pure metal crystal oscillate as the intensity of an applied magnetic field B is increased.Other quantities also oscillate, such as the resistivity (Shubnikov–de Haas effect), specific heat, and soundattenuation. This effect is due to Landau quantization of electron energy in an applied magnetic field. Astrong magnetic field — typically several teslas — and a low temperature are required to cause a materialto exhibit the dHvA effect.

2 BRIEF MATHEMATICAL REVIEW OF HYDRODYNAMICS 4 of 14

2 Brief mathematical review of hydrodynamics

Lets first begin with behavior of ideal2 and inviscid ν := η/ρ ≡ 0 fluid. From Cauchy andPascal law (stress tensor: pik(xl) 6= pik(xl, t) in the absence of a shear forces)

ρui =∂pik∂xk

+ ρ f exi , pik = −pδik (1)

ρDv

Dt≡ ρ

(∂v

∂t+ v · ∇v

)= −∇p+ ρf (2)

where ui is deformation vector, f exi ≡ fi external forces and p hydrostatic preasure. Thisequation is know as Euler equation. If we make assumptions that external forces are conser-vative f = −∇φ, current flow is isotropic and vorticity ω = ∇× v using identity

1

2∇v2 = v × (∇× v) + v ·∇v

0 =Dρ

Dt+ ρ∇ · v

we get Helmholtz equation of vorticity

Dω

Dt− ω

ρ

Dρ

Dt= ρ

D

Dt

(ω

ρ

)= ω ·∇v (3)

This equation is in general too complicated, however, for a start let us look at incompressiblefluid and liquids, where the third dimension is negligible - for 2D fluid flow, we get

Dω

Dt=

(∂ω

∂t+ v ·∇ ω

)= 0 (4)

Interpretation of this equation is that vorticity, when moving with liquid, is conserved - thisis rigorously true only in 2D, which tells Kelvin theorem. We define circulation

Γ :=

∮C(t)

v · dr (5)

where C(t) represents loop moving together with liquid. Using parametrization with naturalparameter s, C(t) : r(s, t), s ∈ [0, 1], r(0, t) = r(1, t), than

Γ(t) =

∮C(t)

v · dr =

∫(∇× v) · dS =

∫ω · dS = const. (6)

this means, if in the irrotational fluid, fluid did not had vorticity at the beginning ∇×v(r, t =0) = 0 - this property conserved also at later time - this is known as potential flow 3. Kelvintheorem is known as vorticity theorem conservation. Let us define another quantity e.i.complex potential (2D)

w(z) = φ(z) + iψ(z), z = x+ iy (7)

2In the deriving the equations of motion we have taken no account of processes of energy dissipation,which may occure in a moving fluid in consequence of internal friction (viscosity) in the fluid and heatexchange between different parts of it - motions of fluids in which thermal conductivity and viscosity areunimportant.

3potential flow beacause v = ∇φ, where φ is velocity potential.

2 BRIEF MATHEMATICAL REVIEW OF HYDRODYNAMICS 5 of 14

where φ velocity potential, ψ stream function. Stream function ψ, is defined by

vx :=∂ψ

∂y

(=∂φ

∂x

)andexplain vy := −∂ψ

∂x

(=∂φ

∂y

)(8)

∇ · v =∂vx∂x

+∂vy∂y

=∂2ψ

∂x∂y− ∂2ψ

∂y∂x

Velocity of vortices in two dimensions (x, y) using Biot-Savart’s law (see [2],[10],[12]) incilindrical geometry

vθ =Γ

2πvr = 0

w(z) = −i Γ

2πln z =

Γ

2πθ + i

−Γ

2πln r = φ+ iψ (9)

Let vortices be at z1 = (x1, y2), . . . , zn = (xn, yn), . . ., than 2D velocity field at this vorticesdistribution has a form

w(z) = − i

2π

∑i

Γiln (z − zi) =∑i

wi(z) (10)

Try to use same principle but now for N vortices, each has its own circulation Γi ≡ κi.Velocity field distribution v(z) := dw(z)

z= vx − ivy and let us this Hamiltonian function

H = − 1

2π

∑i 6=j

κiκjln |zi − zj| (11)

and can be written as

κidxidt

=∂H∂yi

κidyidt

= −∂H∂xi

dHdt

=∂H∂xi

dxidt

+∂H∂yi

dyidt

= 0

For ν := η/ρ 6= 0,∇ · v 6= 0 (η dynamic viscosity, ρ fluid density) we get Navier-Stokesequation

ρDv

Dt= ρf ex −∇p+ η∇2v + (ζ +

1

3η)∇∇ · v (12)

ζ := λ + 23η = const., λ Lame coefficient and η dynamic viscosity. For the final, we define

enstrophy for incompressible fluid ∇ · v = 0 and f = −∇Φ

∂

∂t

∫V

(1

2ρv2 + ρΦ

)d3r = −η

∫V

(∇× v)2 d3r ≥ 0 (13)

When f = 0 it can be seen that viscosity η ≥ 0. However, in incompressible fluid withvortices there is alaways energy dissipation!

3 ONSAGER THEORY OF TURBULENCE 6 of 14

Figure 1: The large, oval-shaped mark on the clouds is the Great Red Spot - believed to be anintense atmospheric disturbance. An anticyclonic vortex in the upper atmosphere of Jupiterhas existed at least since it was observed in 1610 by Galileo with one of the first telescopes -at its widest, it is about three times the diameter of the Earth. Similar large-scale, long-livedvortices exist in the atmospheres of the other gas giant planets of our solar system e.i. “icy”giant Neptune, Saturn etc (Voyager 2, NASA)

3 Onsager theory of turbulence

3.1 Onsager’s theory of point-vortex equilibria

In the published paper Onsager (see [7]) discussed a simple Hamiltonian particle modelof 2D ideal fluid flow, the point-vortex model of Helmholtz and Kirchhoff, describing thismotion for a system of N vortices in a plane, or of straight and paralel line vortices in threedimensions (see [14]). If the planar coordinates of the ith vortex are ri = (xi, yi) and if thatvortex carries a net circulation κi, the equations of motion are

κidxidt

=∂H

∂yiκidyidt

= −∂H∂xi

(14)

H = − 1

2π

∑i<j

κiκj ln(rij/L) (15)

where H is the fluid kinetic energy, rij is the distance between the ith and j th vortex andL is an arbitrary length scale. Where there are no bounderies, H has the form (15).

Modern source for the point-vortex model is Marchioro and Pulvirenti (1994) in which itis provided that the model describes the motion of concentrated blobs of vorticity, evolvingaccording to the 2D incompressible Euler equations, as long as the distance between theblobs is much greater than their diameters4. Another result in the opposite direction states

4In fact, these equations of motion have been formally derived from quatum many-body equations forparellel line vortices in superfluids.

3 ONSAGER THEORY OF TURBULENCE 7 of 14

Figure 2: Great Dark Spot of Neptune is thought to be a hole, similar to the hole in ozonelayer on Earth, however unlike Jupiter’s Great Red Spot, Dark Spot can dissapeare - this wasfirst noticed with disappearance in 1989, in 1994 was replaced by a similar “spot”. Neptune,made up chiefly of hydrogen, helium, water and silicates - does not have a solid surface.Neptune clouds consist mainly of frozen methane. Deep down inside Neptune, the planet“might” have a solid surface (astronomers try to explain dissipative behavior of a Neptunespot)(Voyager 2, NASA).

that a smooth solution of the 2D Euler equations ω(r, t) can be approximated as N → ∞,over any finite time interval, by a sum ωN(r, t) =

∑Ni=1 κiδ(r− ri(t)), where κi = ±1/N and

ri(t), i = 1, . . . , N are the solutions (14).

Theoretical proposal for explanation for a commonly observed feature of nearly two-dimensional flows was: the spontaneous appearance of large-scale, long-lived vortices. Forexamples are the large, lingering storms in the atmospheres of the gas giants of the outersolar system, such as the Great Red Spot of Jupiter (see [16],[17]). Large vortices are alsoreadly seen downstream of flow obstacles. Jupiter is heavier than any other planet. Its massis 318 times larger than that of Earth. Although Jupiter has a large mass, it has relativelylow density (∼ 1.33g/cm3) - about 1/4 of Earth density. Astronomers believe that the planetconsists primarly of hydrogen (∼ 86%) and helium (∼ 14%) (the lighest elements), and tinyamounts of methane, ammonia, phosphine, water, acetylene, ethane, germanium, and car-bon monoxide. Interesting is also that chemicals have formed colorful layers of clouds atdifferent heights (as can be seen from Figure 1 ). White clouds, which are the highest in thezones, are made of crystals of frozen ammonia, darker (lower) clouds are of other chemicalsoccur in the belts - the lowest level are blue clouds. As it can be seen, Great Red Spotcolour differ much from neighboring due to small amounts of sulfur and phosphorus in theammonia crystals.Astronomers began to use telescopes to observe these features in the late 1600’s, the featureshave, however, changed size and brightness but have kept the same patterns.

3 ONSAGER THEORY OF TURBULENCE 8 of 14

Figure 3: CUDA simulation, using Jos Stam’s FFT algorithm, solving Navier-Stokes equationin two-dimensional flow (Nvidia, Open GL Fluid).

Explanation of the phenomenon, Onsager proposed an application of Gibbsian equilibriumstatistical mechanics to the point-vortex model (see [5]). His theory assumed that the gen-eration of the large-scale vortices was a consequence of the inviscid Euler equations, whichform a Hamiltonian system conserving total kinetic energy. This is rigorously true in 2Ddue to conservation of enstrophy. In particular, no sustained forcing is required to maintainthe vortex in this theory as long as the dissipation by viscosity is weak 5 (see [15]). When wecompare our idealized model with reality, we have one profound difference: the distributionsof vorticity which occure in the actual flow of normal fluids are continuous. As a statisticalmodel in two-dimensions it is ambiguous: what set of discrete vortices will best approximatea countinuous distribution of vorticity6?

Burgers in his articles attempts to applay statistical-mechanical maximum entropy ideasto turbulent flow (for non-equilibrium conditions see [9]). The ingenious step in Onsager’stheory was his realization that point vortices would yield states of negative absolute tempera-ture, at sufficiently high energy, and that this result could explain the sponeaneous appearanceof large-scale vortices in two-dimensional flows (see [3], [4], [7]) 7.

5Onsager also assumed the validity of the point-vortex approximation, though with reservations.6Onsager assumed: “...the point-vortex dynamics is ergodic in phase space over the surface of constant

energy, so that a microcanonical distribution is achieved at long times ... We inquire about the ergodicmotion of the system.

7Onsager, in his notes, doesn not explain physical concept of negative-absolute temperature

3 ONSAGER THEORY OF TURBULENCE 9 of 14

Now how to explain negative-absolute temperature? The essential requirements for athermodynamical system to be capable of negative temperature

1. the elements of the thermodynamical system must be in thermodynamical equilibriumamong themselves in order for the system to be described by a temperature at all.

2. there must be an upper limit to the possible energy of the allowed states of the system- negative temperatures are to be achieved with a finite energy.

3. the system must be thermally isolated from all systems which do not satisfy bothupper requirements - internal thermal equilibrium time among the elements of thesystem must be short compared to the time during which appreciable energy is lost toor gained from other systems.

Let us assume that we have a spin-system (most of the system do not satisfy condition(2.),however spin-system can satisfy all three of the conditions)8 where, we add more and moreenergy, temperature starts off positive, approaches positive infinity as maximum entropy isapproached, with half of all spins up. After that, the temperature becomes negative infinite,coming down in magnitude toward zero, but always negative, as the energy increases towardmaximum. When the system has negative temperature, it is hotter than when it is haspositive temperature. If you take two copies of the system, one with positive and one withnegative temperature, and put them in thermal contact, heat will flow from the negative-temperature system into the positive-temperature system. Some systems does not have aproperty that entropy increases monotonically with energy, however are cases when energyis added to the system configuration acctually decreases for some energies (in some energyregion).

How can be this realized in the real world or it is just theoretical science fiction? Bestto explain is using a spin system in a magnetic field. Atoms “must” have other degrees offreedom in addition to spin, making (usually) total energy of the sistem unbounded upwarddue to translation degree of freedom. Sometimes it is useful to define ”spin-temperature“of a collection of atoms but only one condition is met, that is, the coupling between atomicspins and the other degrees of freedom is sufficiently weak, and the coupling between atomicspins sufficiently strong, that the timescale for energy to flow from the spins into otherdegrees of freedom is very large compared to the timescale for thermalization of the spinsamong themselves. Using this condition make sense to talk about temperature of spinsseparately from the temperature of the atoms as a whole - in strong magnetic fields we canmet described condition. Interesting is also that only certain degrees of freedom of a particlecan have negative absolute temperature.

For the existence of negative absolute temperature is the same as that published by theOnsager (1949d) some two years prior to their introduction by Purchell for nuclear-spinsystem (see [1]). Electron (nuclear) spin can be promoted to negative absolute temperaturesby using suitable radio frequency tehniques9.

8there is no upper limit to the possible kinetic energy of a gas molecule9Various experiments and applications (RF amplifier) in the calorimetry of negative temperatures can be

found (see [3],[4])

3 ONSAGER THEORY OF TURBULENCE 10 of 14

The crucial feature of the point-vortex system which permits this conclusion is the factthat the total phase-space volume is finite. Since x and y components of the vortices arecanonically conjugate variables, the total phase-space volume is Φ(∞) = AN , where A is thearea of the flow domain and

Φ(E) =

∫ N∏i=1

d2ri θ(E −H(r1, . . . , rN)) (16)

where θ is the Heviside step function θ(x > 0) = 1 and θ(x < 0) = 0. Φ(E) is a non-negativeincreasing function of energy E, with constant limits Φ(−∞) = 0 and Φ(∞) = AN

Ω(E) = Φ′(E) =

∫ N∏i=1

d2ri δ(E −H(r1, . . . , rN)) (17)

is a non-negative function going to zero at both extremes, Ω(±∞) = 0 . Thus the functionmust achieve a maximum value at some finite Em, where Ω′(Em) = 0 (I use this withreservation!!). For E > Em, Ω′(Em) < 0. On the other hand, by Boltzmann’s principle,the thermodynamic entropy is S(E) = kB lnΩ(E) 10 and the inverse temperature 1/Θ =dS/dE < 0 for E > Em. Onsager further pointed out that negative temperatures will leadto the formation of large-scale vortices by clustering of smaller ones. More precise: in theformer case when 1/Θ > 0, vortices of oposite sign will tend to approach each other. However,if 1/Θ < 0, then vortices of the same sign will tend to cluster - preferably the strongest ones- so as to use up energy at the least possible cost in terms of degree of freedom. It standsto reason that the large compound vortices formed in this matter will remain as the onlyconspicuous features of the motion; because the weaker vortices, free to roam practically atrandom, will yield rather erratic and disorganized contribution to the flow.

The statistical tendency of vortices of the same sign to cluser in the negative-temperatureregime is clear from a description by a canonical distribution∝ e−βH with β = 1/kBΘ. Negative β corresponds to reversing the sign of the interaction,making like “charges” statisticaly attract and opposite “charges” repel.

3.2 Point-vortex model

Montgomery returned to Onsager’s theory and worked out a predictive equation for thelarge-scale vortex solutions (see [8]) conjectured by Onsager11.A brief review of the Joyce-Montgomery considerations, in the language of the 2D point-vortex system. Consider a neutral system, which we describe as consisting of N vortices ofcirculation +1/N and N vortices of circulation −1/N ; for this system, there are two vortexdensities,

ρ±(r) =1

N

N∑i=1

δ(r− r±i ) (18)

10the entropy of a system in which all states, of number Ω, are equally likely; such as system is one inwhich the volume, number of molecules, and internal energy are fixed - the microcanonical ensemble.

11Onsager carried these considerations no further in his 1949 paper nor in any subsequent published work.Von Neumann (1949) took note of the point-vortex model (see [18],[11]) and Onsager’s statistical-mechanicaltheory; this led von N. to speculate about the limits of Kolmogorov’s reasoning in the 3D and to recognizethe profound consequances of enstrophy conservation in two dimensions.

3 ONSAGER THEORY OF TURBULENCE 11 of 14

where r±i , i = 1, . . . , N , are the positions of the N vortices of circulation ±1/N , respectively.Vorticity field is represented by ω(r) := ρ+(r)−ρ−(r). Joyce and Montgomery (1973) derivedformula for the entropy (per particle) of a given field of vortex densities

S = −∫

d2rρ+(r) ln ρ+(r)−∫

d2rρ−(r) ln ρ−(r) (19)

They next reasoned that the equilibrium distributions should be those which maximized theentropy subject to the constraints of fixed energy

E =1

2

∫d2r′G(r, r′)ω(r)ω(r′),

∫d2rρ±(r) ≡ 1 (20)

From here, we work out the variational equation

ρ±(r) = exp

[∓β∫

d2r′G(r, r′)ω(r′)− βµ±]

(21)

where β and µ± are Langrange multipliers to enforce the constraints, having the interpre-tation of inverse temperature and chemical potentials, respectively. A closed equation isobtained by introducing the stream function

ψ(r) =

∫d2r′G(r, r′)ω(r′)

−∆ψ(r) := ω(r) = exp [−β(ψ(r) + µ+)]− exp [β(ψ(r)− µ−)] (22)

This is the final equation derived by Joyce and Montgomery. Its maximum-entropy solutionsgive exact, stable, stationary solutions of the 2D Euler equations and should describe themacroscopic vortices proposed by Onsager when β < 0.

3.3 Advance and applications

One issue that Onsager never addressed was the appropriate thermodynamic limit forvalidity of this statistical theory of large-scale 2D vortices. The Debye-Huckel theory12 isvalid in the standard thermodynamic limit in 2D, for which area A→∞ with the numberof vortices N → ∞ and energy E → ∞ in such a way that n = N/A, e = E/A tend toa finite limit. Further, the circulation κi are held fixed, independent of N , e.g. κi = ±1.The inverse temperature 1/T scales as13 A/N , since

∑i κ

2i ∼ O(N), and approaches a finite

limit in thermodynamic limit. As E/A varies over all real values the temperature T stayspositive, and even more, Campbell and O’Nell (1991) rigorously proved that the standardthermodynamic limit exists for the point-vortex model, but yields only positive temperatures.To obtain the negative temperature states proposed by Onsager, one must consider energiesthat are considerably higher, greater than the critical energy.

Point-vortex approximation: we have already mentioned that there are rigorous resultswhich show that any smooth 2D Euler solution ω(r, t) may be approximated arbitrarily wellover a finite time interval 0 < t < T by a sum of point vortices

∑Ni=1 κiδ(r − ri(t)) with

12Onsager realize the connection with Debye-Huckel theory of electrolytes; that is enough to know for thisseminar.

13shown by Edwards and Taylor (1974).

3 ONSAGER THEORY OF TURBULENCE 12 of 14

κi ∼ ci/N , where ci are constants as N → ∞ (Marchioro and Pulvirenti, 1994)14. Onsagerhad own reservations about the point-vortex approximation - the restrictions imposed by theincompressibility of the fluid. Onsager’s concerns can be cleary understood by consideringthe initial condition of an ideal vortex patch, with a constant level of vorticity on a finitearea. Because that area is conserved by incompressibility under the 2D Euler dynamics, it isnot possible for the vorticity to concentrate or to intensify locally for this initial condition.However, this is not true of one were to approximate the patch by a distribution of pointvortices at high energies. In that case, the mean-square distance between point vorticescould decrease over time and the effective area covered could similarly decrease, leading tomore intense, localized vortex structure. Thus one expects discrepancies here between thecontinuum 2D Euler and the point-vortex model for long times.

A great step toward eliminating these defects was taken independently by Miller (1991)and Robert (1990). They both elaborated an equilibrium statistical-mechanical theory di-rectly for the continuum 2D Euler equations, without making the point-vortex approxima-tion. The basic object of both of these theories was a local distribution function n(r, σ), theprobability density that the microscopic vorticity ω(r) lies between σ and σ+dσ at the spacer. The picture here is that the vorticity field in its evolution mixes to very fine scales sothat a small neighborhood of the point r will contain many values of the vorticity with levelsdistributed according to n(r, σ), thus n satisfies

∫dσn(r, σ) ≡ 1 at each point r in the flow

domain. Same as before, the stream function satisfies the generalized mean-field equation

−∆ψ(r) =1

Z(r)

∫dσ σexp

[−β(σψ(r)− µ(σ))

](23)

where Z(r), µ(σ) and β are Langrange multipliers. This theory is is an application to 2DEuler of the method worked out by Lynden-Bell (1967) to describe gravitional equilibriumafter “violent relaxation” in stellar systems. The Robert-Miller theory solves the problemsdiscussed by Onsager - point-vortex assumption. The new theory incorporates infinitly manyconservation laws of 2D Euler. In fact, the point-vortex model, in the generality consideredby Onsager, also has infinitely many conserved quantities, i.e. the total number of vortices ofa given circulation. Robert-Miller theory includes information about the area of the vorticitylevel sets, which is lacking in the point-vortex model.As rekarked by Miller et al., the Joyce-Montgomery mean-field equation is formally recoveredin a “dilute-vorticity limit” in which the area of the level sets shrinks to zero keeping thenet circulation fixed 15.

Lundgren and Pointin (1976) performed numerical simulations of the point-vortex model(see [1]) with initial conditions corresponding to several local clusters of vortices at somedistance from each other. The equilibrium theory predicted their final coalescence into asingle large supervortex. Lundgren and Pointin argued theoretically that the vortices willeventially reach the equilibrium, single-vortex state.

14this is not sufficient to justify equlibrium statistical mechanics at long times because the limits T →∞and N →∞ need not commute.

15This corresponds well with conditions suggested by Onsager for validity of the point-vortex model that“vorticity is mostly concentrated in small regions.”

4 CONCLUSIONS 13 of 14

4 Conclusions

Onsager observed the spontaneous appearance of large-scale, long lived vortices is a fre-quent occurence in two-dimensional flow, especially in planetary atmospheres. Onsager’stheory, from mathematical point of view, is now largely explored and undestood and alsothose of its genaralization by Miller and Robert (it has been mentioned at the end). As theempirical confirmation of the equilibrium vortex theories is concerned, it must be addmitedthat while reasonable agreement has been obtained with a few numberical simulations andlaboratory experiments (see [13]), we know of not really convincing verification for flows innature. Onsager’s theoretical view of an “ideal turbulence” described by the inviscid fluidequations is a proper idealization for understanding high Reynolds number flows. All ofthe mentioned processes are truly predictive devices especially cascade theory of dissipation.However both theories (Onsager and Kolmogorov) are based on experimental observation,where Kolmogorov’s theory(cascade theory) (see [19]) is based on important hypothesescombined with dimensional arguments. Theory of hydrodynamic turbulence have manyunanswered questions: what are the scales where vorticity is dissipated, or in the otherwords, what is the size of the smallest eddies that are responsible for dissipating the energy?Some are explained by Kolmogorov, but some are still unanswered.

REFERENCES 14 of 14

References

[1] G. L. Eyink, K. R. Sreenivasan, Onsager and the theory of hydrodynamic turbulence (Reviewsof modern physics, Volume 78, January 2006).

[2] L. D. Landau and E. M. Lifshitz, Fluids Mechanics (Pergamon Press, 1987).

[3] N.F. Ramsey, Thermodynamics and statistical mechanics at negative absolute temperature(Phys. Rev. 103, 20 (1956))

[4] M.J. Klein,Negative Absolute Temperature (Phys. Rev. 104, 589 (1956))

[5] G. K. Batchlor, The theory of homogeneous turbulence (Cambridge University Press , 1959).

[6] Onsager, L., 1953a, Proceedings of the International Conference on Theoretical Physics, Kyotoand Tokyo (pp. 669− 675, Sept. 1953).

[7] Onsager, L., private notes, LOA, NTNU, circa 1975.

[8] Parisi, G., and U. Frisch, Turbulence and Predictability in Geopysical Fluid Dynamics, Pro-ceedings of the International School of Physics “E. Fermi” (pp. 84− 87, 1985).

[9] J. Cardy, G. Falkovich, K. Gawedzki, Non-equilibrium Statistical Mechanics and Turbulence(London Mathematical Society LectCambridge,ure Note Series 355, Cambridge UniversityPress, 2008).

[10] R. Robert, Handbook of Mathematical Fluid Dynamics (Elsevier, Amsterdam, Vol. 2, pp.1− 54).

[11] von Neumann J., recent theories of turbulence, reproduced in Collected Works of J. vonNeumann (Pergamon, New York, 1963, pp. 437− 472).

[12] R. Podgornik, Mehanika kontinuuov (skripta, Avgust, 2002).

[13] R. Ecke, The Turbulence Problem, An Experimentalist’s Perspective (Los Alamos Science,Number 29, 2005).

[14] G. Falkovich, K. R. Sreenivasan, Lessons from Hydrodynamic Turbulence (Physics Today,April 2006).

[15] S. Jung, P. J. Morrison, H. Swinney, Statistical mechanics of two-dimensional turbulence (J.Fluid Mech., vol. 554, pp.433-456, Cambridge University Press, 2006).

[16] R. C. Lien, T. B. Sanford, Spectral characteristics of velocity and vorticity fluxes in an un-stratified turbulent boundary layer (Applied Physical Laboratory and School of Oceanography,College of Ocean and Fishery Science, University of Washington, Seattle).

[17] P. B. Weichman, Statistical Mechanics of Fluid Flow and Large-Scale Ocean Circulation (BEASystems, Advanced Inforamtion Tehnologies, Presentation).

[18] Frisch, U., Turbulence: The Legacy of A. N. Kolomogorov (Cambridge and New York, Cam-bridge Univeristy Press, 1995).

[19] Zakharov, V., L’vov, V. and Falkovich, G., Kolmogorov Spectra of Turbulence (Springer, Berlinand New York, 1992).